Systems of Inequalities Characterizing Ring Homomorphisms

Abstract

Assume thatT:P→RandU:P→Rare arbitrary mappings between two partially ordered ringsPandR. We study a few systems of functional inequalities which characterize ring homomorphisms. For example, we prove that ifTandUsatisfyT(f+g)≥T(f)+T(g),  U(f·g)≥U(f)·U(g),for allf,g∈PandT≥U, thenU=Tand this mapping is a ring homomorphism. Moreover, we find two other systems for which we obtain analogous assertions.